Theorem neisspw 13219

Description: The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
neifval.1	|- X = U. J

Assertion

Ref	Expression
neisspw	|- ( J e. Top -> ( ( nei ` J ) ` S ) C_ ~P X )

Proof of Theorem neisspw

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . 5 |- X = U. J
2	1	neii1 13218	. . . 4 |- ( ( J e. Top /\ v e. ( ( nei ` J ) ` S ) ) -> v C_ X )
3		velpw 3579	. . . 4 |- ( v e. ~P X <-> v C_ X )
4	2, 3	sylibr 134	. . 3 |- ( ( J e. Top /\ v e. ( ( nei ` J ) ` S ) ) -> v e. ~P X )
5	4	ex 115	. 2 |- ( J e. Top -> ( v e. ( ( nei ` J ) ` S ) -> v e. ~P X ) )
6	5	ssrdv 3159	1 |- ( J e. Top -> ( ( nei ` J ) ` S ) C_ ~P X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   C_ wss 3127   ~Pcpw 3572   U.cuni 3805   ` cfv 5208   Topctop 13066   neicnei 13209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-top 13067  df-nei 13210

This theorem is referenced by: (None)